Determine the point of intersection for the system of equations: \( y = 3x - 4 \) and \( y = -2x + 8 \)

To find the point of intersection for the system of equations: 1. \( y = 3x - 4 \) 2. \( y = -2x + 8 \) we need to find the values of \( x \) and \( y \) that satisfy both equations simultaneously. ### Step 1: Set the Equations Equal to Each Other Since both equations are equal to \( y \), we can set them equal to each other: \[ 3x - 4 = -2x + 8 \] ### Step 2: Solve for \( x \) First, let's get all the \( x \)-terms on one side and the constant terms on the other side. 1. **Add \( 2x \) to both sides:** \[ 3x + 2x - 4 = 8 \] \[ 5x - 4 = 8 \] 2. **Add \( 4 \) to both sides:** \[ 5x = 12 \] 3. **Divide both sides by \( 5 \):** \[ x = \frac{12}{5} \quad \text{or} \quad x = 2.4 \] ### Step 3: Solve for \( y \) Now that we have the value of \( x \), we can substitute it back into either of the original equations to find \( y \). Let's use the first equation: \[ y = 3x - 4 \] Substitute \( x = \frac{12}{5} \): \[ y = 3\left(\frac{12}{5}\right) - 4 \] \[ y = \frac{36}{5} - \frac{20}{5} \] \[ y = \frac{16}{5} \quad \text{or} \quad y = 3.2 \] ### Step 4: Write the Point of Intersection The point where the two lines intersect is: \[ \left( \frac{12}{5}, \frac{16}{5} \right) \quad \text{or} \quad (2.4, 3.2) \] ### Conclusion The system of equations intersects at the point **\(\left( \frac{12}{5}, \frac{16}{5} \right)\)**, which is equivalent to **\((2.4, 3.2)\)**.